Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elharval | Structured version Visualization version GIF version |
Description: The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
elharval | ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6696 | . 2 ⊢ (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V) | |
2 | reldom 8503 | . . . 4 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5602 | . . 3 ⊢ (𝑌 ≼ 𝑋 → 𝑋 ∈ V) |
4 | 3 | adantl 482 | . 2 ⊢ ((𝑌 ∈ On ∧ 𝑌 ≼ 𝑋) → 𝑋 ∈ V) |
5 | harval 9014 | . . . 4 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) | |
6 | 5 | eleq2d 2895 | . . 3 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})) |
7 | breq1 5060 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ 𝑋 ↔ 𝑌 ≼ 𝑋)) | |
8 | 7 | elrab 3677 | . . 3 ⊢ (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
9 | 6, 8 | syl6bb 288 | . 2 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))) |
10 | 1, 4, 9 | pm5.21nii 380 | 1 ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2105 {crab 3139 Vcvv 3492 class class class wbr 5057 Oncon0 6184 ‘cfv 6348 ≼ cdom 8495 harchar 9008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-dom 8499 df-oi 8962 df-har 9010 |
This theorem is referenced by: harndom 9016 harcard 9395 cardprclem 9396 cardaleph 9503 dfac12lem2 9558 hsmexlem1 9836 pwcfsdom 9993 pwfseqlem5 10073 hargch 10083 harinf 39509 harn0 39580 |
Copyright terms: Public domain | W3C validator |